Next: The electric dipole approximation
Up: Approximation methods
Previous: Harmonic perturbations
Let us use some of the results of timedependent perturbation theory
to investigate the interaction of an atomic electron with
classical (i.e., nonquantized) electromagnetic radiation.
The unperturbed Hamiltonian
is

(824) 
The standard classical prescription for obtaining the Hamiltonian of
a particle
of charge
in the presence of an electromagnetic field is
where
is the vector potential and
is the scalar potential. Note that
This prescription also works in quantum mechanics. Thus, the Hamiltonian
of an atomic electron placed in an electromagnetic field is

(829) 
where and are functions of the position operators.
The above equation can be written

(830) 
Now,

(831) 
provided that we adopt the gauge
.
Hence,

(832) 
Suppose that the perturbation corresponds to a monochromatic planewave, for which
where
and are unit vectors which specify the direction
of polarization and the direction of propagation, respectively.
Note that
. The Hamiltonian
becomes

(835) 
with

(836) 
and

(837) 
where the term, which is second order in , has been neglected.
The perturbing Hamiltonian can be written

(838) 
This has the same form as Eq. (812), provided that

(839) 
It is clear, by analogy with the previous analysis, that the first
term on the righthand side of Eq. (838) describes the absorption
of a photon of energy , whereas the second term describes
the stimulated emission of a photon of energy . It follows from
Eq. (822) that the rate of absorption is

(840) 
The absorption crosssection is defined as the ratio of the
power absorbed by the atom to the incident power
per unit area in the electromagnetic field. Now the energy density of an electromagnetic field
is

(841) 
where and
are the peak electric and magnetic fieldstrengths,
respectively. The incident power per unit area of the electromagnetic field
is

(842) 
Now,

(843) 
so

(844) 
Next: The electric dipole approximation
Up: Approximation methods
Previous: Harmonic perturbations
Richard Fitzpatrick
20060216